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776 n A Textbook of Machine Design Flywheel 776 1. Introduction. 2. Coefficient of Fluctuation of Speed. 3. Fluctuation of Energy. 4. Maximum Fluctuation of Energy. 5. Coefficient of Fluctuation of Energy. 6. Energy Stored in a Flywheel. 7. Stresses in a Flywheel Rim. 8. Stresses in Flywheel Arms. 9. Design of Flywheel Arms. 10. Design of Shaft, Hub and Key. 11. Construction of Flywheel . 22 C H A P T E R 22.122.1 22.122.1 22.1 IntrIntr IntrIntr Intr oductionoduction oductionoduction oduction A flywheel used in machines serves as a reservior which stores energy during the period when the supply of energy is more than the requirement and releases it during the period when the requirement of energy is more than supply. In case of steam engines, internal combustion engines, reciprocating compressors and pumps, the energy is developed during one stroke and the engine is to run for the whole cycle on the energy produced during this one stroke. For example, in I.C. engines, the energy is developed only during power stroke which is much more than the engine load, and no energy is being developed during suction, compression and exhaust strokes in case of four stroke engines and during compression in case of two stroke engines. The excess energy developed during power stroke is absorbed by the flywheel and releases it to the crankshaft during other strokes in which no energy is developed, thus CONTENTS CONTENTS CONTENTS CONTENTS Flywheel n 777 rotating the crankshaft at a uniform speed. A little consideration will show that when the flywheel absorbs energy, its speed increases and when it releases, the speed decreases. Hence a flywheel does not maintain a constant speed, it simply reduces the fluctuation of speed. In machines where the operation is intermittent like punching machines, shearing machines, riveting machines, crushers etc., the flywheel stores energy from the power source during the greater portion of the operating cycle and gives it up during a small period of the cycle. Thus the energy from the power source to the machines is supplied practically at a constant rate throughout the operation. Note: The function of a governor in engine is entirely different from that of a flywheel. It regulates the mean speed of an engine when there are variations in the load, e.g. when the load on the engine increases, it becomes necessary to increase the supply of working fluid. On the other hand, when the load decreases, less working fluid is required. The governor automatically controls the supply of working fluid to the engine with the varying load condition and keeps the mean speed within certain limits. As discussed above, the flywheel does not maintain a constant speed, it simply reduces the fluctuation of speed. In other words, a flywheel controls the speed variations caused by the fluctuation of the engine turning moment during each cycle of operation. It does not control the speed variations caused by the varying load. 22.222.2 22.222.2 22.2 CoefCoef CoefCoef Coef ff ff f icient of Fluctuaicient of Fluctua icient of Fluctuaicient of Fluctua icient of Fluctua tion of Speedtion of Speed tion of Speedtion of Speed tion of Speed The difference between the maximum and minimum speeds during a cycle is called the maximum fluctuation of speed. The ratio of the maximum fluctuation of speed to the mean speed is called coefficient of fluctuation of speed. Let N 1 = Maximum speed in r.p.m. during the cycle, N 2 = Minimum speed in r.p.m. during the cycle, and N = Mean speed in r.p.m. = 12 2 + NN ∴ Coefficient of fluctuation of speed, C S = () 12 12 12 2 NN NN NNN − − = + = () 12 12 12 2 ω−ω ω−ω = ωω+ω (In terms of angular speeds) = () 12 12 12 2 vv vv vvv − − = + (In terms of linear speeds) The coefficient of fluctuation of speed is a limiting factor in the design of flywheel. It varies depending upon the nature of service to which the flywheel is employed. Table 22.1 shows the per- missible values for coefficient of fluctuation of speed for some machines. Note: The reciprocal of coefficient of fluctuation of speed is known as coefficient of steadiness and it is de- noted by m . ∴ m = S121212 1 ω === −ω−ω− Nv CNN vv Flywheel stores energy when the supply is in excess, and releases energy when the supply is in deficit. 778 n A Textbook of Machine Design TT TT T aa aa a ble 22.1.ble 22.1. ble 22.1.ble 22.1. ble 22.1. P P P P P erer erer er missible vmissible v missible vmissible v missible v alues falues f alues falues f alues f or coefor coef or coefor coef or coef ff ff f icient of ficient of f icient of ficient of f icient of f luctualuctua luctualuctua luctua tion of speed (tion of speed ( tion of speed (tion of speed ( tion of speed ( CC CC C SS SS S ).). ).). ). S.No. Type of machine or class of service Coefficient of fluctuation of speed (C S ) 1. Crushing machines 0.200 2. Electrical machines 0.003 3. Electrical machines (direct drive) 0.002 4. Engines with belt transmission 0.030 5. Gear wheel transmission 0.020 6. Hammering machines 0.200 7. Pumping machines 0.03 to 0.05 8. Machine tools 0.030 9. Paper making, textile and weaving machines 0.025 10. Punching, shearing and power presses 0.10 to 0.15 11. Spinning machinery 0.10 to 0.020 12. Rolling mills and mining machines 0.025 22.322.3 22.322.3 22.3 FluctuaFluctua FluctuaFluctua Fluctua tion of Enertion of Ener tion of Enertion of Ener tion of Ener gygy gygy gy The fluctuation of energy may be determined by the turning moment diagram for one complete cycle of operation. Consider a turning moment diagram for a single cylinder double acting steam engine as shown in Fig. 22.1. The vertical ordinate represents the turning moment and the horizontal ordinate (abscissa) represents the crank angle. A little consideration will show that the turning moment is zero when the crank angle is zero. It rises to a maximum value when crank angle reaches 90º and it is again zero when crank angle is 180º. This is shown by the curve abc in Fig. 22.1 and it represents the turning moment diagram for outstroke. The curve cde is the turning moment diagram for instroke and is somewhat similar to the curve abc. Since the work done is the product of the turning moment and the angle turned, therefore the area of the turning moment diagram represents the work done per revolution. In actual practice, the engine is assumed to work against the mean resisting torque, as shown by a horizontal line AF. The height of the ordinate aA represents the mean height of the turning moment diagram. Since it is assumed that the work done by the turning moment per revolution is equal to the work done against the mean resisting torque, therefore the area of the rectangle aA Fe is proportional to the work done against the mean resisting torque. Fig. 22.1. Turning moment diagram for a single cylinder double acting steam engine. We see in Fig. 22.1, that the mean resisting torque line AF cuts the turning moment diagram at points B, C, D and E. When the crank moves from ‘a’ to ‘p’ the work done by the engine is equal to Flywheel n 779 the area aBp, whereas the energy required is represented by the area aABp. In other words, the engine has done less work (equal to the area aAB) than the requirement. This amount of energy is taken from the flywheel and hence the speed of the flywheel decreases. Now the crank moves from p to q, the work done by the engine is equal to the area pBbCq, whereas the requirement of energy is represented by the area pBCq. Therefore the engine has done more work than the requirement. This excess work (equal to the area BbC) is stored in the flywheel and hence the speed of the flywheel increases while the crank moves from p to q. Similarly when the crank moves from q to r, more work is taken from the engine than is developed. This loss of work is represented by the area CcD. To supply this loss, the flywheel gives up some of its energy and thus the speed decreases while the crank moves from q to r. As the crank moves from r to s, excess energy is again developed given by the area DdE and the speed again increases. As the piston moves from s to e, again there is a loss of work and the speed decreases. The variations of energy above and below the mean resisting torque line are called fluctuation of energy. The areas BbC, CcD, DdE etc. represent fluctuations of energy. Fig. 22.2. Tunring moment diagram for a four stroke internal combustion engine. A little consideration will show that the engine has a maximum speed either at q or at s. This is due to the fact that the flywheel absorbs energy while the crank moves from p to q and from r to s. On the other hand, the engine has a minimum speed either at p or at r. The reason is that the flywheel gives out some of its energy when the crank moves from a to p and from q to r. The difference between the maximum and the minimum energies is known as maximum fluctuation of energy. A turning moment diagram for a four stroke internal combustion engine is shown in Fig. 22.2. We know that in a four stroke internal combustion engine, there is one working stroke after the crank has turned through 720º (or 4π radians). Since the pressure inside the engine cylinder is less than the atmospheric pressure during suction stroke, therefore a negative loop is formed as shown in Fig. 22.2. During the compression stroke, the work is done on the gases, therefore a higher negative loop is obtained. In the working stroke, the fuel burns and the gases expand, therefore a large positive loop is formed. During exhaust stroke, the work is done on the gases, therefore a negative loop is obtained. A turning moment diagram for a compound steam engine having three cylinders and the resultant turning moment diagram is shown in Fig. 22.3. The resultant turning moment diagram is the sum of Flywheel shown as a separate part 780 n A Textbook of Machine Design the turning moment diagrams for the three cylinders. It may be noted that the first cylinder is the high pressure cylinder, second cylinder is the intermediate cylinder and the third cylinder is the low pressure cylinder. The cranks, in case of three cylinders are usually placed at 120º to each other. Fig. 22.3. Turning moment diagram for a compound steam engine. 22.422.4 22.422.4 22.4 MaximMaxim MaximMaxim Maxim um Fluctuaum Fluctua um Fluctuaum Fluctua um Fluctua tion of Enertion of Ener tion of Enertion of Ener tion of Ener gygy gygy gy A turning moment diagram for a multi-cylinder engine is shown by a wavy curve in Fig. 22.4. The horizontal line AG represents the mean torque line. Let a 1 , a 3 , a 5 be the areas above the mean torque line and a 2 , a 4 and a 6 be the areas below the mean torque line. These areas represent some quantity of energy which is either added or subtracted from the energy of the moving parts of the engine. Fig. 22.4. Turning moment diagram for a multi-cylinder engine. Let the energy in the flywheel at A = E, then from Fig. 22.4, we have Energy at B = E + a 1 Energy at C = E + a 1 – a 2 Energy at D = E + a 1 – a 2 + a 3 Energy at E = E + a 1 – a 2 + a 3 – a 4 Energy at F = E + a 1 – a 2 + a 3 – a 4 + a 5 Energy at G = E + a 1 – a 2 + a 3 – a 4 + a 5 – a 6 = Energy at A Let us now suppose that the maximum of these energies is at B and minimum at E. ∴ Maximum energy in the flywheel = E + a 1 and minimum energy in the flywheel = E + a 1 – a 2 + a 3 – a 4 Flywheel n 781 ∴ Maximum fluctuation of energy, ∆E = Maximum energy – Minimum energy =(E + a 1 ) – (E + a 1 – a 2 + a 3 – a 4 ) = a 2 – a 3 + a 4 22.522.5 22.522.5 22.5 CoefCoef CoefCoef Coef ff ff f icient of Fluctuaicient of Fluctua icient of Fluctuaicient of Fluctua icient of Fluctua tion of Enertion of Ener tion of Enertion of Ener tion of Ener gygy gygy gy It is defined as the ratio of the maximum fluctuation of energy to the work done per cycle. It is usually denoted by C E . Mathematically, coefficient of fluctuation of energy, C E = Maximum fluctuation of energy Work done per cycle The workdone per cycle may be obtained by using the following relations: 1. Workdone / cycle = T mean × θ where T mean = Mean torque, and θ = Angle turned in radians per revolution =2 π, in case of steam engines and two stroke internal combustion engines. =4 π, in case of four stroke internal combustion engines. The mean torque (T mean ) in N-m may be obtained by using the following relation i.e. T mean = 60 2 PP N × = πω where P = Power transmitted in watts, N = Speed in r.p.m., and ω = Angular speed in rad/s = 2πN /60 2. The workdone per cycle may also be obtained by using the following relation: Workdone / cycle = 60P× n where n = Number of working strokes per minute. = N, in case of steam engines and two stroke internal combustion engines. = N / 2, in case of four stroke internal combustion engines. The following table shows the values of coefficient of fluctuation of energy for steam engines and internal combustion engines. TT TT T aa aa a ble 22.2.ble 22.2. ble 22.2.ble 22.2. ble 22.2. Coef Coef Coef Coef Coef ff ff f icient of ficient of f icient of ficient of f icient of f luctualuctua luctualuctua luctua tion of enertion of ener tion of enertion of ener tion of ener gy (gy ( gy (gy ( gy ( CC CC C EE EE E ) f) f ) f) f ) f or steam and interor steam and inter or steam and interor steam and inter or steam and inter nalnal nalnal nal combustion engines.combustion engines. combustion engines.combustion engines. combustion engines. S.No. Type of engine Coefficient of fluctuation of energy (C E ) 1. Single cylinder, double acting steam engine 0.21 2. Cross-compound steam engine 0.096 3. Single cylinder, single acting, four stroke gas engine 1.93 4. Four cylinder, single acting, four stroke gas engine 0.066 5. Six cylinder, single acting, four stroke gas engine 0.031 22.622.6 22.622.6 22.6 EnerEner EnerEner Ener gy Storgy Stor gy Storgy Stor gy Stor ed in a Flywheeled in a Flywheel ed in a Flywheeled in a Flywheel ed in a Flywheel A flywheel is shown in Fig. 22.5. We have already discussed that when a flywheel absorbs energy its speed increases and when it gives up energy its speed decreases. 782 n A Textbook of Machine Design Fig. 22.5. Flywheel. Let m = Mass of the flywheel in kg, k = Radius of gyration of the flywheel in metres, I = Mass moment of inertia of the flywheel about the axis of rotation in kg-m 2 = m.k 2 , N 1 and N 2 = Maximum and minimum speeds during the cycle in r.p.m., ω 1 and ω 2 = Maximum and minimum angular speeds during the cycle in rad / s, N = Mean speed during the cycle in r.p.m. = 12 , 2 NN + ω = Mean angular speed during the cycle in rad / s = 12 2 ω+ω C S = Coefficient of fluctuation of speed = 12 − NN N or 12 ω−ω ω We know that mean kinetic energy of the flywheel, E = 222 11 22 Imk ×ω= × ω (in N-m or joules) As the speed of the flywheel changes from ω 1 to ω 2 , the maximum fluctuation of energy, ∆E = Maximum K.E. — Minimum K.E. = 22 12 11 () () 22 ×ω −×ω II = 1 2 × I 22 12 1212 1 () () ( )( ) 2 I  ω−ω =×ω+ωω−ω  = I.ω (ω 1 – ω 2 ) 12 2 ω+ω  ω=   ∵ (i) = I.ω 2 12 ω−ω   ω  [Multiplying and dividing by ω] = I.ω 2 .C S = m.k 2 .ω 2 .C S ( ∵ I = m.k 2 ) (ii) =2 E.C S 2 1 . 2  =×ω   ∵ EI (iii) The radius of gyration (k) may be taken equal to the mean radius of the rim (R), because the thickness of rim is very small as compared to the diameter of rim. Therefore substituting k = R in equation (ii), we have ∆ E = m.R 2 .ω 2 .C S = m.v 2 .C S ( ∵ v = ω.R ) From this expression, the mass of the flywheel rim may be determined. Notes: 1. In the above expression, only the mass moment of inertia of the rim is considered and the mass moment of inertia of the hub and arms is neglected. This is due to the fact that the major portion of weight of the flywheel is in the rim and a small portion is in the hub and arms. Also the hub and arms are nearer to the axis of rotation, therefore the moment of inertia of the hub and arms is very small. 2. The density of cast iron may be taken as 7260 kg / m 3 and for cast steel, it may taken as 7800 kg / m 3 . 3. The mass of the flywheel rim is given by m = Volume × Density = 2 π R × A × ρ Flywheel n 783 From this expression, we may find the value of the cross-sectional area of the rim. Assuming the cross-section of the rim to be rectangular, then A = b × t where b = Width of the rim, and t = Thickness of the rim. Knowing the ratio of b /t which is usually taken as 2, we may find the width and thickness of rim. 4. When the flywheel is to be used as a pulley, then the width of rim should be taken 20 to 40 mm greater than the width of belt. Example 22.1. The turning moment diagram for a petrol engine is drawn to the following scales: Turning moment, 1 mm = 5 N-m; Crank angle, 1 mm = 1º. The turning moment diagram repeats itself at every half revolution of the engine and the areas above and below the mean turning moment line, taken in order are 295, 685, 40, 340, 960, 270 mm 2 . Determine the mass of 300 mm diameter flywheel rim when the coefficient of fluctuation of speed is 0.3% and the engine runs at 1800 r.p.m. Also determine the cross-section of the rim when the width of the rim is twice of thickness. Assume density of rim material as 7250 kg / m 3 . Solution. Given : D = 300 mm or R = 150 mm = 0.15 m ; C S = 0.3% = 0.003 ; N = 1800 r.p.m. or ω = 2 π × 1800 / 60 = 188.5 rad/s ; ρ = 7250 kg / m 3 Mass of the flywheel Let m = Mass of the flywheel in kg. First of all, let us find the maximum fluctuation of energy. The turning moment diagram is shown in Fig. 22.6. Since the scale of turning moment is 1 mm = 5 N-m, and scale of the crank angle is 1 mm = 1° = π / 180 rad, therefore 1 mm 2 on the turning moment diagram. = 5 × π / 180 = 0.087 N-m Let the total energy at A = E. Therefore from Fig. 22.6, we find that Energy at B = E + 295 Energy at C = E + 295 – 685 = E – 390 Energy at D = E – 390 + 40 = E – 350 Energy at E = E – 350 – 340 = E – 690 Energy at F = E – 690 + 960 = E + 270 Energy at G = E + 270 – 270 = E = Energy at A From above we see that the energy is maximum at B and minimum at E. ∴ Maximum energy = E + 295 and minimum energy = E – 690 784 n A Textbook of Machine Design We know that maximum fluctuation of energy, ∆ E = Maximum energy — Minimum energy =(E + 295) – (E – 690) = 985 mm 2 = 985 × 0.087 = 86 N-m We also know that maximum fluctuation of energy (∆ E), 86 = m.R 2 .ω 2 .C S = m (0.15) 2 (188.5) 2 (0.003) = 2.4 m ∴ m = 86 / 2.4 = 35.8 kg Ans. Fig. 22.6 Cross-section of the flywheel rim Let t = Thickness of rim in metres, and b = Width of rim in metres = 2 t (Given) ∴ Cross-sectional area of rim, A = b × t = 2 t × t = 2 t 2 We know that mass of the flywheel rim (m), 35.8 = A × 2πR × ρ = 2t 2 × 2π × 0.15 × 7250 = 13 668 t 2 ∴ t 2 = 35.8 / 13 668 = 0.0026 or t = 0.051 m = 51 mm Ans. and b =2 t = 2 × 51 = 102 mm Ans. Example 22.2. The intercepted areas between the output torque curve and the mean resistance line of a turning moment diagram for a multicylinder engine, taken in order from one end are as follows: – 35, + 410, – 285, + 325, – 335, + 260, – 365, + 285, – 260 mm 2 . The diagram has been drawn to a scale of 1 mm = 70 N-m and 1 mm = 4.5°. The engine speed is 900 r.p.m. and the fluctuation in speed is not to exceed 2% of the mean speed. Find the mass and cross-section of the flywheel rim having 650 mm mean diameter. The density of the material of the flywheel may be taken as 7200 kg / m 3 . The rim is rectangular with the width 2 times the thickness. Neglect effect of arms, etc. Solution. Given : N = 900 r.p.m. or ω = 2π × 900 / 60 = 94.26 rad/s ; ω 1 – ω 2 = 2% ω or 12 ω−ω ω = C S = 2% = 0.02 ; D = 650 mm or R = 325 mm = 0.325 m ; ρ = 7200 kg / m 3 Mass of the flywheel rim Let m = Mass of the flywheel rim in kg. First of all, let us find the maximum fluctuation of energy. The turning moment diagram for a multi-cylinder engine is shown in Fig. 22.7. Since the scale of turning moment is 1 mm = 70 N-m and scale of the crank angle is 1 mm = 4.5º = π / 40 rad, therefore 1 mm 2 on the turning moment diagram. = 70 × π / 40 = 5.5 N-m Flywheel n 785 Fig. 22.7 Let the total energy at A = E. Therefore from Fig. 22.7, we find that Energy at B = E – 35 Energy at C = E – 35 + 410 = E + 375 Energy at D = E + 375 – 285 = E + 90 Energy at E = E + 90 + 325 = E + 415 Energy at F = E + 415 – 335 = E + 80 Energy at G = E + 80 + 260 = E + 340 Energy at H = E + 340 – 365 = E – 25 Energy at K = E – 25 + 285 = E + 260 Energy at L = E + 260 – 260 = E = Energy at A From above, we see that the energy is maximum at E and minimum at B. ∴ Maximum energy = E + 415 and minimum energy = E – 35 We know that maximum fluctuation of energy, =(E + 415) – (E – 35) = 450 mm 2 = 450 × 5.5 = 2475 N-m We also know that maximum fluctuation of energy (∆E), 2475 = m.R 2 .ω 2 .C S = m (0.325) 2 (94.26) 2 0.02 = 18.77 m ∴ m = 2475 / 18.77 = 132 kg Ans. Cross-section of the flywheel rim Let t = Thickness of the rim in metres, and b = Width of the rim in metres = 2 t (Given) ∴ Area of cross-section of the rim, A = b × t = 2 t × t = 2 t 2 We know that mass of the flywheel rim (m), 132 = A × 2 π R × ρ = 2 t 2 × 2 π × 0.325 × 7200 = 29 409 t 2 ∴ t 2 = 132 / 29 409 = 0.0044 or t = 0.067 m = 67 mm Ans. and b =2t = 2 × 67 = 134 mm Ans. Example 22.3. A single cylinder double acting steam engine develops 150 kW at a mean speed of 80 r.p.m. The coefficient of fluctuation of energy is 0.1 and the fluctuation of speed is ± 2% of mean speed. If the mean diameter of the flywheel rim is 2 metres and the hub and spokes provide 5 percent of the rotational inertia of the wheel, find the mass of the flywheel and cross-sectional area of the rim. Assume the density of the flywheel material (which is cast iron) as 7200 kg / m 3 .